In any field of nonlinear analysis Nemytskij operators, the superposition operators generated by appropriate functions, play a crucial part. Their analytic properties depend on the postulated properties of the defining function and on the function space in which they are considered. A rich source for related questions is the monograph by J. Appell and P. P. Zabrejko [2] and the survey paper by J. Appell [1].

Canonically associated to a real symplectic vector space are several associative algebras. The Weyl algebra (generated by the Heisenberg commutation relations) has been the subject of much study; see [1] for example. The exponential Weyl algebra (generated by the canonical commutation relations in exponential form) has been less well studied; see [8].

Let A be a Banach algebra with unit 1 and let B be a Banach algebra which is a subalgebra of A and which contains 1. In [5]Barnes gave sufficient conditions for B to be inverse closed in A. In this paper we consider single elements and study the question of how the spectrum relative to B of an element in B relates to the spectrum of the element relative to A.

If G1 and G2 are locally compact groups and the algebras Ll(G1) and Ll(G2) are isometrically isomorphic, then G1 and G2 are isomorphic (Wendel, 1952, [8]). There is evidence that the following generalization of Wendel's result is true.

If T is an algebra isomorphism of L1(G1) onto L1(G2) with ∥T∥ < √2, then G1, and G2 are isomorphic.

In this paper we characterize semigroups S which have a semigroup Q of left quotients, where Q is an ℛ-unipotent semigroup which is a band of groups. Recall that an ℛ-unipotent (or left inverse) semigroup S is one in which every ℛ-class contains a unique idempotent. It is well-known that any ℛ-unipotent semigroup 5 is a regular semigroup in which the set of idempotents is a left regular band in that efe = ef for any idempotents e, fin S. ℛ-unipotent semigroups were studied by several authors, see for example [1] and [13].Bailes [1]characterized ℛ-unipotent semigroups which are bands of groups. This characterization extended the structure of inverse semigroups which are semilattices of groups. Recently, Gould studied in [7]the semigroup S which has a semigroup Q of left quotients where Q is an inverse semigroup which is a semilattice of groups.

In [2] I described the deformation of bending a hyperbolic manifold along an embedded totally geodesic hypersurface. As I remarked there, the deformation is particularly interesting in the case of a surface, because a surface contains many embedded totally geodesic hypersurfaces, namely simple closed curves, along which it is possible to bend. Furthermore, for a surface it is possible to extend the definition of bending to the case of a geodesic lamination, by using the fact that the set of simple closed geodesies is dense in the space of geodesic laminations. This direction has been developed by Epstein and Marden in [1].

A Banach space E is said to have Mazur's property if every weak* sequentially continuous functional in E” is weak* continuous, i.e. belongs to E. Such spaces were investigated in [5] and [9] where they were called d-complete and μB-spaces respectively. The class of Banach spaces with Mazur's property includes the WCG spaces and, more generally, the Banach spaces with weak* angelic dual balls [4, p. 564]. Also, it is easy to see that Mazur's property is inherited by closed subspaces. The main goal of this paper is to present generalizations of some results of [5] concerning the stability of Mazur's property with respect to forming some tensor products of Banach spaces. In particular, we show in Sections 2 and 3 that the spaces E ⊗εF and Lp(μ, E) inherit Mazur's property from E andF under some conditions. In Section 4, we will also show the stability of Mazur's property under the formation of Schauder decompositions and some unconditional sums of Banach spaces.

Let (S, ≤) be a poset (partially ordered set), A(S) = Aut(S, ≤) its automorphism group and G ⊆ A(S) a subgroup. In the literature, various authors have studied sufficient conditions on G and the structure of (S, ≤) which imply that G is simple or perfect. Let us call (S, ≤) doubly homogeneous if each isomorphism between two 2-subsets of 5 extends to an isomorphism of (S, ≤). Higman [8] proved that if (S, ≤) is a doubly homogeneous chain then B(S), the group of all automorphisms of (S, ≤) with bounded support, is simple, and each element of B(S) is a commutator in B(S). Droste, Holland and Macpherson [5] showed that if (S, ≤) is a doubly homogeneous tree then its automorphism group again contains a unique simple normal subgroup in which each element is a commutator. Dlab [3] established similar results for various groups of locally linear automorphisms of the reals. Further results in this direction are contained in Glass [7]. It is the aim of this note to establish a common generalization and sharpening of the previously mentioned results.

A compact bordered Klein surface X of algebraic genus g ≥ 2 has maximal symmetry [6] if its automorphism group A(X) is of order 12(g — 1), the largest possible. The bordered surfaces with maximal symmetry are clearly of special interest and have been studied in several recent papers ([6] and [9] among others).

Since the pioneering work of W. G. Bade [3, 4] a great deal of work has been done on bounded Boolean algebras of projections on a Banach space ([11, XVII.3.XVIII.3], [21, V.3], [16], [6], [12], [13], [14], ]17], [18], [23], [24]). Via the Stone representation space of the Boolean algebra, the theory can be studied through Banach modules over C(K), where K is a compact Hausdorff space. One of the key concepts in the theory is the notion of Bade functionals. If X is a Banach C(K)-module and x ε X, then a Bade functional of x with respect to C(K) is a continuous linear functional α on X such that, for each a in C(K) with a ≥ 0, we have

(i) α (ax) ≥0,

(ii) if α (ax) = 0, then ax = 0.

An almost Hermitian manifold (, J, g) with Riemannian connection is called nearly Kaehlerian if (xJ)X = 0 for any . The typical example is the sphere S6. The nearly Kaehlerian structure J for S6 is constructed in a natural way by making use of Cayley division algebra [3]. It is because of this nearly Kaehler, non-Kaehler, structure that S6 has attracted attention. Different classes of submanifolds of S6 have been considered by A. Gray [4], K. Sekigawa [5] and N. Ejiri [2]. In this paper we study 2-dimensional totally real submanifolds of S6. These are submanifolds with the property that for every x є M, J (Tx M) belongs to the normal bundle v. For this class we have obtained the following result.

A well-known theorem of E. Posner [10] states that if the composition d1d2 of derivations d1d2 of a prime ring A of characteristic not 2 is a derivation, then either d1 = 0 or d2 = 0. A number of authors have generalized this theorem in several ways (see e.g. [1], [2], and [5], where further references can be found). Under stronger assumptions when A is the algebra of all bounded linear operators on a Banach space (resp. Hilbert space), Posner's theorem was reproved in [3] (resp. [12]). Recently, M. Mathieu [8] extended Posner's theorem to arbitrary C*-algebras.

Much of the research done by different authors on the lattice of kernel functors (equivalently, linear topologies) has been summarized by Golan in [2]. More recently, the rings whose lattices of kernel functors are linearly ordered were introduced in [3] as a categorical generalization of valuation rings in the non-commutative case. Results (and examples) in [3] show that there is an abundance of non-commutative rings R whose lattices (R), both in Mod-R and R-Mod, are simultaneously linearly ordered; however, the question of the symmetry of this condition remained open. Here we will prove that, for every natural number n≥3, there exists a ring Rn such that (Mod-Rn) is a linearly ordered lattice of n elements, whereas (Rn-Mod) is not linearly ordered.

In this note we observe how results of Gordon, Luecke, Feustel, and Whitten establish the precise extent to which knowledge of the knot group determines the knot, in both the oriented and the unoriented case. This enables us to say, in terms of the knot group, exactly when two spun knots are the same.

A generalized tensor product of groups was introduced by R. Brown and J.-L. Loday [6], and has led to a substantial algebraic theory contained essentially in the following papers: [6, 7, 1, 5, 11, 12, 13, 14, 18, 19, 20, 23, 24] ([9, 27, 28] also contain results related to the theory, but are independent of Brown and Loday's work). It is clear that one should be able to develop an analogous theory of tensor products for other algebraic structures such as Lie algebras or commutative algebras. However to do so, many non-obvious algebraic identities need to be verified, and various topological proofs (which exist only in the group case) have to be replaced by purely algebraic ones. The work involved is sufficiently non-trivial to make it interesting.

1. Point (3) of the main theorem of our paper [3, Theorem 1.1] is incorrect: this note corrects the main and consequential errors, and shows that (after minor adjustments) almost all the other results of [3], including the remaining seven points of Theorem 1.1, remain correct.

2. The theme of [3] was a family of functors G,(–), defined on the category of rings with unity for each cardinal t. For t = 0, 1, the results of [3] are unchanged, but, for 2≤t<∞, major, and, for t infinite, less major, corrections are necessary; we therefore assume 2≤t. Terminology and notation are standard or as in [3], and I would like to thank A. W. Chatters and an anonymous referee for comments which prompted this correction.